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Chapter 13 Experimental Design and Analysis of Variance

Chapter 13 Experimental Design and Analysis of Variance . An Introduction to Experimental Design and Analysis of Variance . Analysis of Variance and the Completely Randomized Design. Multiple Comparison Procedures. An Introduction to Experimental Design

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Chapter 13 Experimental Design and Analysis of Variance

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  1. Chapter 13 Experimental Design and Analysis of Variance • An Introduction to Experimental Design and Analysis of Variance • Analysis of Variance and the Completely Randomized Design • Multiple Comparison Procedures

  2. An Introduction to Experimental Design and Analysis of Variance • Statistical studies can be classified as being either experimental or observational. • In an experimental study, one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest. • In an observational study, no attempt is made to control the factors. • Cause-and-effect relationships are easier to establish in experimental studies than in observational studies. • Analysis of variance (ANOVA) can be used to analyze the data obtained from experimental or observational studies.

  3. An Introduction to Experimental Design and Analysis of Variance • In this chapter three types of experimental designs are introduced. • a completely randomized design • a randomized block design • a factorial experiment

  4. An Introduction to Experimental Design and Analysis of Variance • A factor is a variable that the experimenter has selected for investigation. • A treatment is a level of a factor. • Experimental unitsare the objects of interest in the experiment. • A completely randomized designis an experimental design in which the treatments are randomly assigned to the experimental units.

  5. Analysis of Variance: A Conceptual Overview Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means. Data obtained from observational or experimental studies can be used for the analysis. We want to use the sample results to test the following hypotheses: H0: 1=2=3=. . . = k Ha: Not all population means are equal

  6. Analysis of Variance: A Conceptual Overview H0: 1=2=3=. . . = k Ha: Not all population means are equal If H0 is rejected, we cannot conclude that all population means are different. Rejecting H0 means that at least two population means have different values.

  7. Analysis of Variance: A Conceptual Overview • Assumptions for Analysis of Variance For each population, the response (dependent) variable is normally distributed. The variance of the response variable, denoted  2, is the same for all of the populations. The observations must be independent.

  8. Sampling Distribution of Given H0 is True Analysis of Variance: A Conceptual Overview Sample means are close together because there is only one sampling distribution when H0 is true. 

  9. Sampling Distribution of Given H0 is False Analysis of Variance: A Conceptual Overview Sample means come from different sampling distributions and are not as close together when H0 is false. 3 1 2

  10. Analysis of Variance and the Completely Randomized Design • Between-Treatments Estimate of Population Variance • Within-Treatments Estimate of Population Variance • Comparing the Variance Estimates: The F Test • ANOVA Table

  11. Between-Treatments Estimate of Population Variance s 2 • The estimate of  2 based on the variation of the sample means is called the mean square due to treatments and is denoted by MSTR. Numerator is called the sum of squares due to treatments(SSTR) Denominator is the degrees of freedom associated with SSTR

  12. Within-Treatments Estimateof Population Variance s 2 • The estimate of  2 based on the variation of the • sample observations within each sample is called the • mean square error and is denoted by MSE. Numerator is called the sum of squares due to error(SSE) Denominator is the degrees of freedom associated with SSE

  13. Comparing the Variance Estimates: The F Test • If the null hypothesis is true and the ANOVA • assumptions are valid, the sampling distribution of • MSTR/MSE is an F distribution with MSTR d.f. • equal to k - 1 and MSE d.f. equal to nT - k. • If the means of the k populations are not equal, the • value of MSTR/MSE will be inflated because MSTR • overestimates  2. • Hence, we will reject H0 if the resulting value of • MSTR/MSE appears to be too large to have been • selected at random from the appropriate F • distribution.

  14. Comparing the Variance Estimates: The F Test • Sampling Distribution of MSTR/MSE Sampling Distribution of MSTR/MSE Reject H0 a Do Not Reject H0 MSTR/MSE F Critical Value

  15. ANOVA Table for a Completely Randomized Design p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments k - 1 SSTR Error SSE nT - k Total SST nT - 1 SST’s degrees of freedom (d.f.) are partitioned into SSTR’s d.f. and SSE’s d.f. SST is partitioned into SSTR and SSE.

  16. ANOVA Table for a Completely Randomized Design SST divided by its degrees of freedom nT – 1 is the overall sample variance that would be obtained if we treated the entire set of observations as one data set. With the entire data set as one sample, the formula for computing the total sum of squares, SST, is:

  17. ANOVA Table for a Completely Randomized Design ANOVA can be viewed as the process of partitioning the total sum of squares and the degrees of freedom into their corresponding sources: treatments and error. Dividing the sum of squares by the appropriate degrees of freedom provides the variance estimates and the F value used to test the hypothesis of equal population means.

  18. Test for the Equality of k Population Means • Hypotheses H0: 1=2=3=. . . = k Ha: Not all population means are equal • Test Statistic F = MSTR/MSE

  19. Test for the Equality of k Population Means • Rejection Rule Reject H0 if p-value <a p-value Approach: Critical Value Approach: Reject H0 if F>Fa where the value of F is based on an F distribution with k - 1 numerator d.f. and nT - k denominator d.f.

  20. Testing for the Equality of k Population Means: A Completely Randomized Design • Example: AutoShine, Inc. AutoShine, Inc. is considering marketing a long- lasting car wax. Three different waxes (Type 1, Type 2, and Type 3) have been developed. In order to test the durability of these waxes, 5 new cars were waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs of deterioration.

  21. Testing for the Equality of k Population Means: A Completely Randomized Design • Example: AutoShine, Inc. The number of times each car went through the carwash before its wax deteriorated is shown on the next slide. AutoShine, Inc. must decide which wax to market. Are the three waxes equally effective? Factor . . . Car wax Treatments . . . Type I, Type 2, Type 3 Experimental units . . . Cars Response variable . . . Number of washes

  22. Testing for the Equality of k Population Means: A Completely Randomized Design Wax Type 1 Wax Type 2 Wax Type 3 Observation 1 2 3 4 5 27 30 29 28 31 33 28 31 30 30 29 28 30 32 31 Sample Mean 29.0 30.4 30.0 Sample Variance 2.5 3.3 2.5

  23. Testing for the Equality of k Population Means: A Completely Randomized Design • Hypotheses H0: 1=2=3 Ha: Not all the means are equal where: 1 = mean number of washes using Type 1 wax 2 = mean number of washes using Type 2 wax 3 = mean number of washes using Type 3 wax

  24. = (29 + 30.4 + 30)/3 = 29.8 Testing for the Equality of k Population Means: A Completely Randomized Design • Mean Square Between Treatments Because the sample sizes are all equal: SSTR = 5(29–29.8)2 + 5(30.4–29.8)2 + 5(30–29.8)2 = 5.2 MSTR = 5.2/(3 - 1) = 2.6 • Mean Square Error SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2 MSE = 33.2/(15 - 3) = 2.77

  25. Testing for the Equality of k Population Means: A Completely Randomized Design • Rejection Rule p-Value Approach: Reject H0 if p-value < .05 Critical Value Approach: Reject H0 if F> 3.89 where F.05 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom

  26. Testing for the Equality of k Population Means: A Completely Randomized Design • Test Statistic F = MSTR/MSE = 2.60/2.77 = .939 • Conclusion The p-value is greater than .10, where F = 2.81. (Excel provides a p-value of .42.) Therefore, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

  27. Testing for the Equality of k Population Means: A Completely Randomized Design • ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares p-Value F 5.2 .42 Treatments 2 2.60 .939 Error 33.2 12 2.77 Total 38.4 14

  28. Testing for the Equality of k Population Means: An Observational Study • Example: Reed Manufacturing Janet Reed would like to know if there is any significant difference in the mean number of hours worked per week for the department managers at her three manufacturing plants (in Buffalo, Pittsburgh, and Detroit). An F test will be conducted using a = .05.

  29. Testing for the Equality of k Population Means: An Observational Study • Example: Reed Manufacturing A simple random sample of five managers from each of the three plants was taken and the number of hours worked by each manager in the previous week is shown on the next slide. Factor . . . Manufacturing plant Treatments . . . Buffalo, Pittsburgh, Detroit Experimental units . . . Managers Response variable . . . Number of hours worked

  30. Testing for the Equality of k Population Means: An Observational Study Plant 3 Detroit Plant 2 Pittsburgh Plant 1 Buffalo Observation 1 2 3 4 5 48 54 57 54 62 51 63 61 54 56 73 63 66 64 74 Sample Mean 55 68 57 Sample Variance 26.0 26.5 24.5

  31. Testing for the Equality of k Population Means: An Observational Study • p -Value and Critical Value Approaches 1. Develop the hypotheses. H0:  1= 2= 3 Ha: Not all the means are equal where:  1 = mean number of hours worked per week by the managers at Plant 1  2 = mean number of hours worked per week by the managers at Plant 2   3 = mean number of hours worked per week by the managers at Plant 3

  32. = (55 + 68 + 57)/3 = 60 Testing for the Equality of k Population Means: An Observational Study • p -Value and Critical Value Approaches a = .05 2. Specify the level of significance. 3. Compute the value of the test statistic. Mean Square Due to Treatments (Sample sizes are all equal.) SSTR = 5(55 - 60)2 + 5(68 - 60)2 + 5(57 - 60)2 = 490 MSTR = 490/(3 - 1) = 245

  33. Testing for the Equality of k Population Means: An Observational Study • p -Value and Critical Value Approaches 3. Compute the value of the test statistic. (con’t.) Mean Square Due to Error SSE = 4(26.0) + 4(26.5) + 4(24.5) = 308 MSE = 308/(15 - 3) = 25.667 F = MSTR/MSE = 245/25.667 = 9.55

  34. Testing for the Equality of k Population Means: An Observational Study • ANOVA Table Source of Variation Sum of Squares Mean Square Degrees of Freedom F p-Value 245 25.667 9.55 .0033 490 308 798 2 12 14 Treatment Error Total

  35. Testing for the Equality of k Population Means: An Observational Study • p –Value Approach 4. Compute the p –value. With 2 numerator d.f. and 12 denominator d.f., the p-value is .01 for F = 6.93. Therefore, the p-value is less than .01 for F = 9.55. 5. Determine whether to reject H0. The p-value < .05,so we reject H0. We have sufficient evidence to conclude that the mean number of hours worked per week by department managers is not the same at all 3 plant.

  36. Testing for the Equality of k Population Means: An Observational Study • Critical Value Approach 4. Determine the critical value and rejection rule. Based on an F distribution with 2 numerator d.f. and 12 denominator d.f., F.05 = 3.89. Reject H0 if F> 3.89 5. Determine whether to reject H0. Because F = 9.55 > 3.89, we reject H0. We have sufficient evidence to conclude that the mean number of hours worked per week by department managers is not the same at all 3 plant.

  37. Multiple Comparison Procedures • Suppose that analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. • Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

  38. Reject H0 if > LSD Fisher’s LSD ProcedureBased on the Test Statistic xi - xj _ _ • Hypotheses • Test Statistic • Rejection Rule where

  39. Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • Example: Reed Manufacturing Recall that Janet Reed wants to know if there is any significant difference in the mean number of hours worked per week for the department managers at her three manufacturing plants. Analysis of variance has provided statistical evidence to reject the null hypothesis of equal population means. Fisher’s least significant difference (LSD) procedure can be used to determine where the differences occur.

  40. Fisher’s LSD ProcedureBased on the Test Statistic xi - xj For  = .05 and nT - k = 15 – 3 = 12 degrees of freedom, t.025 = 2.179 MSE value was computed earlier

  41. Reject H0 if > 6.98 = |55 - 68| = 13 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 1 and 2 • Hypotheses (A) • Rejection Rule • Test Statistic • Conclusion • The mean number of hours worked at Plant 1 is • not equal to the mean number worked at Plant 2.

  42. Reject H0 if > 6.98 = |55 - 57| = 2 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 1 and 3 • Hypotheses (B) • Rejection Rule • Test Statistic • Conclusion There is no significant difference between the mean number of hours worked at Plant 1 and the mean number of hours worked at Plant 3.

  43. Reject H0 if > 6.98 = |68 - 57| = 11 Fisher’s LSD ProcedureBased on the Test Statistic xi - xj • LSD for Plants 2 and 3 • Hypotheses (C) • Rejection Rule • Test Statistic • Conclusion The mean number of hours worked at Plant 2 is not equal to the mean number worked at Plant 3.

  44. Type I Error Rates • The comparison-wise Type I error ratea indicates the level of significance associated with a single pairwise comparison. • The experiment-wise Type I error rate gets larger for problems with more populations (larger k). • The experiment-wise Type I error rateaEW is the probability of making a Type I error on at least one of the (k – 1)! pairwise comparisons. aEW = 1 – (1 – a)(k – 1)!

  45. Randomized Block Design • Experimental unitsare the objects of interest in the experiment. • A completely randomized designis an experimental design in which the treatments are randomly assigned to the experimental units. • If the experimental units are heterogeneous, blocking can be used to form homogeneous groups, resulting in a randomized block design.

  46. Randomized Block Design • ANOVA Procedure • For a randomized block design the sum of squares total (SST) is partitioned into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error. SST = SSTR + SSBL + SSE • The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term.

  47. Randomized Block Design • ANOVA Table p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments k - 1 SSTR Blocks SSBL b - 1 Error SSE (k – 1)(b – 1) Total SST nT - 1

  48. Randomized Block Design • Example: Crescent Oil Co. Crescent Oil has developed three new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends.

  49. Randomized Block Design • Example: Crescent Oil Co. Five automobiles have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide. Factor . . . Gasoline blend Treatments . . . Blend X, Blend Y, Blend Z Blocks . . . Automobiles Response variable . . . Miles per gallon

  50. Randomized Block Design Type of Gasoline (Treatment) Automobile (Block) Block Means Blend X Blend Y Blend Z 1 2 3 4 5 31 30 29 33 26 30 29 29 31 25 30 29 28 29 26 30.333 29.333 28.667 31.000 25.667 Treatment Means 29.8 28.8 28.4

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